proof of Jacobi's identity for $\vartheta$ functions

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We start with the Fourier transform of $f(x)=e^{{i\pi\tau x^{2}+2ixz}}$ :

\int_{{-\infty}}^{{+\infty}}e^{{i\pi\tau x^{2}+2ixz}}e^{{2\pi ixy}}\,dx=(-i% \tau)^{{-1/2}}e^{{-i{(z+\pi y)^{2}\over\pi\tau}}}

Applying the Poisson summation formula, we obtain the following:

\sum_{{n=-\infty}}^{{+\infty}}e^{{i\pi\tau n^{2}+2inz}}=(-i\tau)^{{-1/2}}\sum_% {{n=-\infty}}^{{+\infty}}e^{{-i{(z+\pi n)^{2}\over\pi\tau}}}

The left hand side equals $\vartheta_{3}(z\mid\tau)$ . The right hand side can be rewritten as follows:

\sum_{{n=-\infty}}^{{+\infty}}e^{{-i{(z+\pi n)^{2}\over\pi\tau}}}=e^{{-i{z^{2}% \over\pi\tau}}}\sum_{{n=-\infty}}^{{+\infty}}e^{{-i{\pi n^{2}\over\tau}-{2inz% \over\tau}}}=e^{{-i{z^{2}\over\pi\tau}}}\vartheta_{3}(z/\tau\mid-1/\tau)

Combining the two expressions yields

\vartheta_{3}(z\mid\tau)=e^{{-i{z^{2}\over\pi\tau}}}\vartheta_{3}(z/\tau\mid-1% /\tau)

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